Additive-multiplicative magic squares, 10th-order and above


After additive-multiplicative magic squares of orders 8-9 (also called addition-multiplication magic squares, or even shorter "add-mult"), what about bigger squares?

Here is his best add-mult magic square of order 10:

And here is an example of my add-mult magic squares:


Table of best known add-mult magic squares

1   

2         3         4
Photos 1 and 2 (kindly provided by A. D. Keedwell) : Jószef Dénes (Budapest 1932 - 2002) with Paul Erdös, summer 1990.
Photo 3 : A. Donald Keedwell (London 1928 - ) in 1979,
Photo 4 : Paul Erdös (Budapest 1913 - Warsaw 1996) in 1992.

"Problem 6.3. For what orders n do addition-multiplication magic squares exist?"
Jószef Dénes and A. Donald Keedwell, Latin Squares and their Applications (1974), page 489.

"Many unsolved problems are stated, some classical, some due to the authors, and even some proposed by the writer of this foreword."
Paul Erdös, foreword of the above Dénes - Keedwell book (1974) , page 5.

In this Dénes - Keedwell book, Horner's 8x8 and 9x9 add-mult squares are published on pages 215-216 and the above quoted problem 6.3 is posed. I do not have the full answer to this Dénes - Keedwell - Erdös problem, but here is a partial answer with a summary of the best known add-mult magic squares that you can download with the Excel file below. When their book was published, only orders 8 and 9 were known. We can note below that orders p (where p is a prime) are difficult to get, but not impossible, as proved by Toshihiro Shirakawa with p = 11, 13, 17, 19.


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